Upon arrival at Singapore Budget Terminal, you'll need to clear customs, collect your luggage, take the free shuttle to the main terminal, and check in for your onward flight with Jetstar.

Even if you went straight for your Jetstar flight, you would have entered Singapore, and not in 'transit' in the usual sense of the word. So your Jetstar flight is considered as a separate flight originating from Singapore.

That's my very long-winded way of saying, 'You need to pay the airport tax.' ;-)